That depends on your definitions. If you use the classic definition of photon as the thingy with the E = \hbar \omega, then no, its energy is fixed, it cannot be amplified.

If you use a popular definition of a singe photon (it's so cute for a photon to be single) as the light produced in a single act of emission, then yes, this beast is amplifiable, but it is not the above thingy.

Dear Igor,

Rightly, i should be clearer about the definition of what is to be amplified. Can you increase the electric field associated with the photon and in so doing reproduce any phase associated with the photon?

So as far as the electric fields are concerned that of the amplified photon is just a larger amplitude of the original single photon.

many thanks,

Neil

A single photon can be amplified, i.e. increase its energy in the process of reradiation.

Dear Neil,

The photon is fundamental particle with a well defined energy and momentum having rest mass zero. Thus if you have it as a single isolated photon in vacuum there are not form to be amplified.

But if you do scatter it with atoms the things can change. If scatters with a single isolated atom, the atom cannot do anything to amplifier it as in vacuum.However if the atom is interacting with other atoms in a solid, there are several mechanisms for radiationless decay i.e. transferring the energy of the absorbed photon into channels that don't involve reradiating the photon. The energy can be transferred to lattice vibrations, i.e, heat, which is generally known as quenching. The fluerescence or the phosporescenced are two effects related with forbidden transition states. The amplification is only possible if this photon coincides with one excited state under very determined situation, but their energy balance is involving losses besides the photons. Thus this is not a good amplifier of the photon too. Perhaps you could reach it if the photon could stimulate the emittion of radiation as in a laser o in a maser, but again the state of bosons (photons) need to be able to decay releasing new coherent photons.

In a gas the excited atom after absorving the photon or molecule can collide with another atom/molecule and transfer the excitation energy into kinetic energy but never to amplifier the initial photon. This is known as collisional de-excitation.

In any case the photon can be absorverd by matter but never gain anergy if that were not on the excited atoms or molecules.

If we stipulate that "amplification of a single photon" implies that "one photon in, and one photon of the same frequency is out", then NO amplification of a single photon is possible (simply ask yourself what is "photon" and what is its energy). However, if you change anything in the above condition, then you can get whatever you want (e. g. a regular stimulated emission, whereby after interaction with an atom, two photons of the same frequency are out -- you stimulated the atom to give away stored energy).

However, if you want to stick with "a single photon out", you can change the condition of "the same frequency", i. e. allow for the photon's frequency/energy to change (in particular -- increase, or "amplify") . For that, let the photon be reflected from a moving mirror -- in this case, due to Doppler effect , its frequency (and thus energy) will go up, if the mirror runs in the direction opposite to that of the incoming photon (the extra energy is borrowed then from the mirror). You can do it many times, and this significantly increase the photon's energy -- there is a whole literature on jacking up photon's energy in a resonator with collapsing mirrors/walls....

Dear Daniel and Alexander,

thank you for your good points.

Just to extend this, consider the thought experiment where a pump creates an entangled pair, then these two photons are mixed together to create a single photon at the sum frequency (ie up conversion). So that's like entanglement followed by the reverse of entanglement. According to momentum conservation, the phase of the pump equals the sum of the phases of the signal and idler, and this then equals the phase of the up converted signal. I know the phase of a single photon makes little sense. However, if i made a long term average of the difference between the phase of the up-converted photon and the pump, would this integrate to a stationary quantity? Classically it should, but since we are dealing with individual photons, what would be the result then?

If you consider the phase difference above would integrate to a stationary quantity, what would happen if you amplified the signal and idler photons by a little stimulated emission. Does this not produce a second photon that is in phase with the first one? The simple answer is that stimulated emission breaks the entanglement, so there would be no coherence, ie would not integrate to a stationary quantity. However, if the phase is preserved through the simulated emission, then perhaps it would integrate to a stationary quantity?

Thank you for any comments.

Neil

As others have said, a single photon has a fixed energy and so cannot be amplified and still be a single photon at the original frequency. However, I think most of what you are interested in is easily understandable if you stop thinking of a single photon.

You say "I know the phase of a single photon makes little sense". Not true. A single photon has a well defined phase which will be apparent if you mix it with another photon.

You wanted to know if a single photon can be amplified. That is exactly what happens in stimulated emission within a laser amplifier. When they say another photon is emitted "in lock step" they mean the emitted photon has the same phase as the original photon. In terms of electric field, the electric field grows in amplitude proportional to the growing photon density and the phase is preserved.

How well the phase is preserved is all about simple physical stuff: slightly different wavelengths due to the slightly different thermal energies of the emitting atoms, dispersion for the slightly different wavelengths, slightly different path lengths for slightly different allowed paths, etc.

The equations for mixing are largely in terms of the electric field without reference to photons, and phase is definitely preserved. In fact phase matching is the critical trick for all mixing processes. The nonlinear materials have dispersion so the different colors move at different speeds. That is equivalent to saying they accumulate phase at different rates. The trick is finding a material where this polarization of that wavelength travelling at that slight angle to another wavelength at the same or different polarization results in a mixed frequency at another slight angle or different polarization all waves accumulating phase at the same rate along one axis.

Dear Mike,

good answer! An alternative to thinking about a single photon would be to consider a wave description. I'm used to classical description of waves expj(wt - k.r + phase), but then how does this look if i'm trying to describe the single photon? The application that i'm considering for this must preserve the information contained within the photon, therefore some how this must be described mathematically so i can determine the effects of eg amplification. Any ideas how this is done?

many thanks,

Neil

For example coherent detection such as can be found in coherent lidar can be single photon sensitive. The signal to be detected is mixed with a coherent local oscillator (usually picked off from the same laser that generates the signal). The mixed signal is then detected on a sensitive receiver. Optical detectors detect power, so that is intrinsically proportional to the square of the field. This constitutes homodyne detection. The strength of the local oscillator raises the signal well above the detector noise level, and a single photon of signal can be detected.

alternatively single photon sensitivity can be achieved by direct optical amplification. Passing the signal through a gain medium results in stimulated emission which amplified the signal preserving all characteristics of phase, frequency, and, if you are careful, polarization

Dear Mike,

that sounds promising. In addition to homodyne, if you take an avalanche photo diode this also "amplifies" the photon to enable its detection. So in some ways, yes the photon can be amplified, but I’m just trying to figure what the penalties will be.

Taking this a little further, if you have two entangled photons (a signal photon having a phase phi_s and an idler photon with a phase phi_i) and amplify both of these separately, say using stimulated emission in a laser, will the entangled photons retain their phases through the amplification process?

I know classically when you amplify a wave, then phase is retained, and the amplitude just increases. However, what might be the quantum description of this process?

Thank you for your comments.

Neil

Yes the phase will be preserved. There are some small caveats.

Stimulated emission can add photons at slightly different wavelengths depending on the gain bandwidth. This means that they will get out of phase as they propagate. The amplified signal has a coherence length which indicates how far the beam can propagate before the phase no longer maintains a fixed relation to the initial photon. However the coherence length after a single pass amplifier can be kilometers.

You also are clearly interested in parametric conversion (although that seems to be not directly part of this question). In the conversion crystal the phase relation between the 4 waves slips. The beams emerge with a fixed phase relation (coherent), but not necessarily the same phase. As I mentioned before much of the art of parametric conversion is finding a way to keep the phases together for as long as possible, but that is never perfect.

Dear Mike,

you make some good points, indicating there is perhaps hope.

Taking this a stage further, if you you replaced the stimulated emission amplifiers with electronic amplifiers, by just coming down in frequency to around a few GHz, would there still be photon phase preservation? Classical electronic amplifiers preserve the phase well. So the question may be, how well is photon phase preserve in electronic amplifiers compared to stimulated emission amplifiers?

(there is less uncertainly in the creation of entangled photons at these lower frequencies, as you just copy into the microwave what's done in the optical)

thank you for your help.

Neil

In mixing down to a lower frequency (heterodyne) the phase of the beat frequency is related to the phase of the components as you expect from your electronics experience.

Alternatively you can just measure the phase and carry that forward. This is done by splitting the signal (so you would first have to amplify your single photon through some gain) and homodyning against a local oscillator AND against the same local oscillator with a 90 degree phase shift. Take the results to be the real and imaginary part of the wave.

Sorry for the multiple copies. When I submitted my phone kept saying there was an error and I didn't look to see that it actually posted anyway. I've deleted the extra copies. I hope I didn't cause any heartburn.

Dear Mike,

Hmmm..., you make it sound all too easy, this explanation sounds a bit too classical, i'm sure there are some 'quantum catches' somewhere; i'm trying to figure where these are.

but many thanks for your enthusiasm.

Neil

You might want to look up articles on coherent LIDAR. Single photon sensitive heterodyne detection is standard practice for that purpose.

many thanks, i'll check this out, cheers, Neil

I think all of the answers so far make good points. Having a background in coherent lidar research, I especially appreciate Mike Albert's answers and comments.

The only thing I would add is that since all particles propagate as probability waves and the quanta only exist as measurement events, all of the mathematics of the classical wave equation for the eletromagnetic (EM) fields apply to photon propagation and interference. The magnitude square of the resultant field gives the probability distribution of detecting a photon event in a measurement.

This is why for photons, the classical EM wave equations can be used to predict the spatial interference pattern in the double slit experiment even when the photon rate is so low that only a single photon at a time would have passed through the slits if light were conceived of as propagating as a stream of discrete photons, which is a misconception, but a widely held misconception.

Similarly, the temporal interference pattern at the detector in homodyne/heterodyne detection is also determined by the classical EM wave equation even when the photon rates of both the signal and local oscillator photons are so low that only one photon would be present at the detector at a time.

Therefore, one answer to Neil's question is that any amplification of the magnitude squared of the EM wave at a point in spacetime amplifies the probability of detecting a photon event at that point in spacetime.

Photons being discrete quanta of the EM field, the probable number and frequency of the photons detected at that spacetime point is determined by the frequency distribution of the energy of the field at that point.

The probable phases of the photons are given by the frequency distribution of the phase of the EM field (i.e., the phase spectrum of the EM field). The phase will have some fundamental randomness due to the uncertainty principle as well as any additional noise due to "environmental" noise sources.

Due to quantum uncertainty, the fields and their spectra must be formulated stochastically.

Dear Brian,

thank you for you good points. From work by Caves it seems you can amplify a photon, or rather the electric disturbances created by the phonon, using a phase preserving amplifier, but you're not increasing the amount of information and you do add a small amount of noise.

best wishes,

Neil

Neil,

Are you referring to Carlton Caves 1982 article "Quantum limits on noise in linear amplifiers?" If so, he points out in that article that there is a difference in noise between phase sensitive and phase insensitive noise amplification.

The homodyne/heterodyne detection referred to earlier by Mike Albert and myself is usually done in a phase insensitive manner, but there are phase sensitive homodyne/heterodyne techniques that show reduced quantum noise. See Article Quantum noise in phase-sensitive heterodyne detection with a...

and Article Quantum theory of phase-sensitive heterodyne detection

Caves also had an article on the use of squeezed states in laser interferometers: Article Quantum-mechanical noise in an interferometer

With squeezed states you can choose to have lower amplitude (i.e., lower photon number) noise or lower phase noise, but at the expense of higher noise of the other type. See: https://www.rp-photonics.com/squeezed_states_of_light.html

If it is phase that you want to measure with lower than the usual quantum limited noise, this article discusses phase super-sensitivity: Article Improving the phase super-sensitivity of squeezing-assisted ...

This article discusses the use of balanced heterodyned detection with quadrature squeezed light: https://arxiv.org/pdf/1112.3155.pdf

This article reviews the application of squeezed light in interferometers:

https://arxiv.org/pdf/1611.03986.pdf

Regards,

Brian

Dear Brian, i'm primarily interested in phase insensitive amplifiers and the 1982 paper, but these references you provide will be very useful for me. Many thanks for those, best wishes, Neil